ISOMETRICAA GEOMETRICAL INTRODUCTION TO

PLANAR CRYSTALLOGRAPHIC GROUPS

in memory of Professor Rinaldo Prisco

Fig. 28 Intersecting glide lines

[Arthur L. Loeb, Color and Symmetry (Wiley 1971, Krieger 1978)]

GEORGE BALOGLOUSUNY OSWEGO, 2007

ISBN 978 -0 -9792076 -0 -0

Œti d‚ semnÓ Ô ·rmonºa kaÁ ue¡øn ti kaÁ m™ga,|Aristot™lhq ∏ Plåtvnoq taytÁ l™gei "Ô d‚ ·rmonºa ®stÁnoªranºa tÓn f¥sin ‘xoysa ueºan kaÁ kalÓn kaÁ daimonºan?tetramerÓq d‚ tÎ dynåmei pefyky¡a d¥o mesøthtaq ‘xeiΩriumhtik¸n te kaÁ ·rmonik¸n, faºnetaº te tÅ m™rh aªt∂qkaÁ tÅ meg™uh kaÁ a ÊperoxaÁ kat' ΩriumØn kaÁ ˝̋̋̋ssssoooommmmeeeettttrrrrººººaaaannnn????®n gÅr dysÁ tetraxørdoiq Wyumºzetai tÅ m™rh"

Aristotle (?), Fragmenta Varia 47 (Pseudo-Plutarchus, De Musica)

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i

INTRODUCTION

Symmetry is a vast subject, significant in art and nature.

Mathematics lies at its root, and it would be hard to find a

better one on which to demonstrate the working of the

mathematical intellect. I hope I have not completely failed in

giving you an indication of its many ramifications, and in

leading you up the ladder from intuitive concepts to abstract

ideas. – Herman Weyl, Symmetry (Princeton, 1952)

Why and how Isometrica, and who would read it?

Back in Spring 1995, one of my SUNY Oswego students submitted the following

one-sentence teacher evaluation: "The course was relatively easy until chapter

11 when I felt that the instructor was as lost as the students"! Chapter 11 --

typically associated with bankruptcy in the so-called 'real world' -- was in that

case the symmetry chapter in Tannenbaum & Arnold's Excursions in Modern

Mathematics: I had casually picked it as one of two 'optional' chapters in my

section of MAT 102 (SUNY Oswego's main General Education course for non-

science majors, consisting of various mathematical topics).

Perhaps that anonymous student's not entirely unjustified comment was the best

explanation for my decision to volunteer to teach MAT 103, a General Education

course devoted entirely to Symmetry, in Fall 1995: better yet, curiosity killed the

cat -- once I started teaching MAT 103 I never took a break from it, gradually

abandoning my passion for rigor and computation in favor of intuition and

visuality.

But where had MAT 103 come from? Following a January 1991 MAA minicourse

ii

(Symmetry Analysis of Repeated Patterns) by Donald Crowe at the San

Francisco Joint Mathematics Meetings, my colleague Margaret Groman

developed (Fall 1992) a new course (Symmetry and Culture) in response to

our General Education Board's call for courses fulfilling the newly introduced

Human Diversity requirement: after all, was Professor Groman not an algebraist

keenly interested in applications of Abstract Algebra (to symmetry for example),

and had Professor Crowe not co-authored a book with anthropologist Dorothy

Washburn titled Symmetries of Culture (Univ. of Washington Press, 1988)?

MAT 103 ceased to fulfill the Human Diversity requirement and was renamed

Symmetries in Spring 1998, but it remained quite popular among non-science

majors as a course fulfilling their Mathematics requirement; it also attracts a few

Mathematics majors now and then. At about the same time I set out (initially in

collaboration with Margaret Groman) to write a book -- not the least because

Washburn & Crowe had temporarily gone out of print -- that was essentially

completed in three stages: January 1999 (chapters 1-5), January 2000 (chapter

6), and August 2001 (chapters 7 & 8). Various projects and circumstances

delayed !official" completion until November 20, 2006 (the day a new computer

forcefully arrived), with the first six chapters posted on my MAT 103 web site

(http://www.oswego.edu/~baloglou/103) as of Fall 2003. In spite of my endless

proofreading and numerous small changes, what you see here is very close in

both spirit and content to the August 2001 version. [For the record, I have only

added 'review' section 6.18 and subsections 1.5.3 & 4.17.4, and also added or

substantially altered figures 4.73, 5.36, 6.121, 6.131, 7.44, and 8.3.]

My initial intent was to write a student-oriented book, a text that our MAT 103

students -- and, why not, students and also !general" readers elsewhere -- would

enjoy and use: this is why it has been written in such unconventional style, and in

the second person in particular; in a different direction, this is why it relies on

minimal Euclidean Geometry rather than Abstract Algebra. Looking now at the

iii

finished product, I can clearly see a partial failure: the absence of exercises and

other frills (available to considerable extent through the MAT 103 web site),

together with an abundance of detail (also spilling into the MAT 103 web site),

may have conspired toward turning a perceived student's book into a teacher's

book. Beyond students and teachers, and despite its humble origins, there may

also be some specialists interested in Isometrica: I will attempt to address these

three plausible audiences in considerable detail below; you may wish to skip

these three sections at first reading and proceed to the end of the Introduction.

Comments for students and general readers

What is this book about, and how accessible is it?

Donald Crowe's 'repeated patterns', better known nowadays as frieze/border

patterns and wallpaper patterns, may certainly be viewed as one of the very first

mathematical (even if accidentally so) creations of humankind: long before they

were recognized as the poor relatives of the three-dimensional structures so dear

to modern scientists, these planar crystallographic groups were being discovered

again and again by repetition/symmetry-seeking native artists in every corner of

the world. This book's goal is therefore the gradual unveiling of the structural and

the mathematical that hides behind the visual and the artistic: so chapters 2 - 4,

and even chapters 5 and 6, are more eye-pleasing than mind-boggling, while

chapters 7 and 8 certainly require more of the reader's attention. It is fair to say

that a determined reader can read the entire book relying only on some high

school mathematics.

iv

Why is Chapter 1 here to begin with?

Good question: this is the only chapter with some algebra (read analytic

geometry) in a heavily geometrical book! The simple answer is that the General

Education Committee of SUNY Oswego would not approve [Spring 1998] a

mathematical course without some mathematical formulas in it... And it took me a

while to come up with a constructive/creative way of incorporating some formulas

into MAT 103, simply by providing an analytical description -- and, quite

unintentionally, classification -- of the four planar isometries (that is, the four

possible types of distance-preserving transformations of the plane).

So, if you are not algebraically inclined, don't hesitate to skip chapter 1 at first

reading: the four planar isometries are indirectly reintroduced in the much more

reader-friendly chapter 2, save for the general rotation, as well in chapters 3 and

4. (At the other end, some readers may be interested only in chapter 1, which is,

I hope, a very accessible and engaging introduction to planar isometries, relying

on neither matrices nor complex numbers.)

Any other reading tips, dear professor?

I have no illusions: most of you are going to merely browse through my book,

even if you happen to be a student whose GPA depends on it... Well, save for the

potentially attractive figures, this book is not browser-friendly: its conversational

style may be tiring to some, and the absence of 'summary boxes' depressing to

others; and let's not forget a favorite student's remark to the effect that "it is odd

that in a book titled Isometrica there is no definition of isometry"! But those

figures are there, slightly over one per page on the average, and most of them

are interesting at worst and seductive at best (me thinks): so start by looking at

appealing figures, then read comments related to them, then read stuff related to

v

those comments, and ... before you know it you will have read everything! After

all, this book talks to you -- are you willing to listen? (My thanks to another

former student for this 'talking-to-me book' comment!)

Why is there no bibliography?

Both because Isometrica is totally self-contained and because suggestions for

further reading are always made in the text (including this introduction) and in

context. Moreover, Washburn & Crowe provides a rather comprehensive

bibliography to which I would have little to add... But if you ask me for one book

that you could or should read before mine, I would not hesitate to recommend

Peter Stevens' Handbook of Regular Patterns (MIT Press, 1981): that is any

math-phobic's dream book and, although I follow it in neither its 'kaleidoscopic'

approach nor its 'multicultural' focus, several figures from Stevens have been

included in Isometrica (with publisher"s permission) as a tribute.

What is there for the non-mathematically inclined?

Despite the inclusion of patterns from Stevens, my book -- as well as MAT 103

in both its present and past forms -- fails to address in depth the cultural aspects

of those patterns and the 'inner motives' of the native artists who created them:

nothing like Paulus Gerdes' Geometry From Africa (Mathematical Association

of America, 1999) or Washburn and Crowe's second book (with updated

bibliography), Symmetry Comes of Age (Univ. of Washington Press, 2004).

Still, I must mention a telling incident: a former student made once a deal with a

quilt maker friend of hers involving the exchange of her copy of Isometrica for a

quilt right after the MAT 103 final exam! In other words, mathematically oriented

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as it happens to be, Isometrica and its 'abstract' designs can still be a source of

inspiration for many non-mathematically inclined readers.

Is Isometrica related to the work of Escher?

Yes and no: Escher's symmetrical drawings, for which he is well known, are

certainly special cases of wallpaper patterns, which are Isometrica's main

focus; but Escher's main achievement, the tiling of the plane by repeated 'real

world' figures, is not discussed at all. Still, it is safe to say that those intrigued by

Escher's creations are likely to be interested in Isometrica; conversely,

Isometrica might be a solid introduction toward a serious reading of Doris

Schattschneider's classic M. C. Escher: Visions of Symmetry (Abrams, 2004).

More generally, Isometrica is not a good source for tilings of any kind; a few

obvious planar tilings are used as standard examples, but there is no mention of

hyperbolic or spherical tilings, and likewise no discussion of Penrose and other

aperiodic (non-repeating) tilings. Still, the curious reader may find Isometrica to

be a good starting point for such topics. (The same applies to other 'popular',

loosely related topics like fractals.)

How about Alhambra?

Granada's famed Moorish palace complex that inspired Escher is barely

mentioned in Isometrica. For a detailed discussion of Alhambra's wallpaper

aspects I would strongly recommend John Jaworski's A Mathematician's Guide

to the Alhambra, currently available through the Jaworski Travel Diaries at

http://www.grout.demon.co.uk/Travel/travel.htm.

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Is Isometrica history-oriented at all?

No. Consistent with the absence of bibliography, any discussion of the subject's

historical development is absent from Isometrica. For such information, and a

broader view as well, the interested reader is referred to both the internet and

such classics as Grunbaum & Shephard's Tilings and Patterns (Freeman,

1987) and Coxeter's Introduction to Geometry (Wiley, 1980).

Comments for teachers.

Symmetry as a General Education course?

This is an eminently legitimate concern: is it fair for a course that for most of its

takers is their 'final' mathematical experience to be devoted to a single subject

almost devoid of 'real world' applications? My response is that students may in

the end understand more about what Mathematics is about by focusing on one

subject and its development than by being briefly exposed to a variety of

subjects. (Besides, even if I wrote Isometrica for a General Education course, it

may certainly be used for other classes and audiences!)

Is Symmetry just about border and wallpaper patterns?

Certainly not! In fact MAT 103 does cover the isometries of the cube and the

viii

soccerball (and their compositions) toward the end, and students tend to enjoy

these subjects at least as much as the rest of the course (especially when it

comes to isometry composition, which is now greatly facilitated by finiteness). It

is therefore fair to say that Isometrica may also be used for only part of a course

devoted to symmetry or geometry; for example, one may spend just three to four

weeks covering only chapters 2, 3, and 4, or merely two weeks on chapters 2

(border patterns) and 4 (wallpaper patterns).

What is the interplay between border patterns and wallpaper patterns?

Border patterns are planar designs invariant under translation in precisely one

direction; wallpaper patterns are planar designs invariant under translation in two,

therefore infinitely many, directions. This difference makes border patterns

substantially easier to understand and classify. It is therefore natural to use

border patterns as a stepping stone to wallpaper patterns. Further, border

patterns may be seen as the building blocks of wallpaper patterns, and this is

indeed an opportunity that Isometrica does not pass by; the subject is treated in

depth in Shredded Wallpaper -- Bonita Bryson's 2005 honors thesis currently

available at http://www.oswego.edu/~baloglou/103/bryson-thesis.pdf, which

may also be used as a quick introduction to border and wallpaper patterns.

How about covering border patterns only?

I would discourage this option, except perhaps early in high school, with the

intention of covering wallpaper patterns the year after. I suspect nonetheless that

several readers of Isometrica may limit their serious reading to chapter 2, which

is probably the book's most successful and accessible chapter anyway!

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How do border and wallpaper patterns relate to Euclidean Geometry?

The Euclidean Geometry employed in Isometrica is so minimal and elementary

that a daring question emerges: would it actually be possible to develop the

students' geometrical intuition through some informal exposure to border and

wallpaper patterns before introducing them to Euclidean Geometry? Could the

intense exposure to shapes and transformations enforced by the study of

patterns facilitate the absorption of geometrical ideas and even arguments

encountered in high school geometry? This might be a good research topic for

Mathematics educators.

Could this be too easy for some students?

Yes, especially in case they happen to be visual learners. It is the teacher"s

responsibility to decide whether his/her students would benefit from a course

based either partly or wholly on Isometrica, and how much time should be spent

on it (if any). I have seen students who struggled for a D in MAT 103, as well as

students who stated that it was the easiest course (in any subject) they have ever

taken! Anyway, I do suspect that Isometrica could keep even the very best

Mathematics/Science majors intrigued for a weekend (or at least a long Saturday

afternoon), so please do not automatically give up on it simply because you

happen to teach the best and brightest… [And do not forget that student"s

comment at the beginning of this Introduction – it can be a treacherous subject!]

What is the role of color?

The coverage of two-colored patterns in chapters 5 (border patterns) and 6

x

(wallpaper patterns) is a direct consequence of Isometrica's debt to Washburn

& Crowe already alluded to. But, while for Washburn and Crowe the study of the

artistically/anthropologically important two-colored patterns was an end, for me it

ended up being largely a mean: indeed a careful look at chapters 5 and 6 shows

how the classification of two-colored patterns is largely used as an excuse to

delve into the structure of (one-colored) border and wallpaper patterns, and the

compositions of their isometries in particular.

Is Isometrica written top-down or bottom-up?

The answer lies hidden in the previous paragraph! Assuming that it would be

difficult for (my) students to understand first 'abstract' (even if geometrically

presented) composition of isometries (as treated in chapter 7) and then pattern

structure based on that (top-down approach), I opted for an indirect, if not

surreptitious, introduction to isometry composition departing from various

classification issues in chapters 5 and 6 (bottom-up approach). My assumption is

a questionable one, so a student-friendly top-down approach may indeed be

presented in a future book! (In fact such an approach is currently being tested in

Patterns and Transformations (MAT 203), an experimental SUNY Oswego

course for honors students.)

What is the significance of isometry composition?

Finding the isometries of any given pattern is a great exercise for the student,

and essential for the pattern's correct classification. But it is not possible to

appreciate a pattern's structure and 'personality' without understanding the way

its isometries interact with each other: any two pattern isometries combined --

that is, applied sequentially -- produce a third isometry that also leaves the

xi

pattern invariant; it is for this reason that mathematicians talk about border/frieze

and wallpaper groups, the total absence of Group Theory from Isometrica

notwithstanding.

As already indicated, chapter 7 offers a thorough coverage of isometry

composition in a totally geometrical context -- perhaps the most thorough (as well

as accessible) coverage of compositions of planar isometries to be found in any

book. It is therefore possible to use chapter 7 for a largely self-contained (despite

the references to pattern structure) introduction to planar isometry composition.

At the other end, section 7.0 alone shows how isometry composition can be

studied 'empirically' in the context of multi-colored symmetrical tilings: that is in

fact the way isometry composition is studied [since Spring 1997] in MAT 103,

definitely making for the hardest part of the course -- likened once to "pulling

teeth" by one of my best students! (To make 'isometry hunting' more fun, the

instructor may even choose to initially hide from the students the helpful fact that,

when it comes to isometry composition, rotations/translations and (glide)

reflections act like positive and negative numbers in multiplication, respectively.)

What is the significance of isometry recovery?

Finding the isometries of a border pattern is quite easy for most students.

Wallpaper patterns are a different story, complicated by more than one possible

direction for glide reflection, rotations other than half turn, etc. As indicated in

passing in chapter 4, the determination of all the isometries mapping a

'symmetrical' set to a copy of it -- a 'recovery' process discussed in detail in

chapter 3 -- can make the isometries of a complex wallpaper pattern much more

visible and 'natural': quite often the isometries mapping a 'unit' of the pattern to a

copy of it are extendable to the entire pattern! This is stressed in MAT 103:

students are initially encouraged to reconstruct the isometries, with the hope (or

xii

rather certainty) that they will gradually become more capable of seeing them;

they are in fact told that "what you cannot see you may build", a guiding

principle throughout the course! (A student's mother was thrilled enough by this

principle to tell her daughter "now I do know that you are learning something in

college" -- a very sweet comment indeed.) So, even though chapter 4 is almost

entirely independent of chapter 3, I am strongly in favor of covering both.

How do students benefit from classifying patterns?

A former student told me once that "this course put some order in his mind"; and

several students report in their evaluations that MAT 103 made them better

thinkers. For such a visual, almost playful, course these comments may appear

startling at first. But the classification process, especially of two-colored patterns,

is very much a thinking process; for example, and very consistently with the

guiding principle cited above, the classifier will often either detect or rule out an

isometry based on logical rather than visual evidence.

What is the role of symmetry plans?

Washburn & Crowe facilitates the classification of individual two-colored

patterns by way of step-by-step, question-and-answer flow charts; Isometrica

reaches this goal through a complete graphic description of each two-colored

type's isometries and their effect on color (preserving or reversing). This

approach has the advantage of constantly and constructively exposing the

students to the full isometry structure of the 7 border patterns (through 24 two-

colored types and symmetry plans at the end of chapter 5) and the 17 wallpaper

patterns (through 63 two-colored types and symmetry plans at the end of chapter

6). Quite clearly, similar symmetry plans could be used for the simpler tasks of

xiii

classifying one-colored border patterns (chapter 2) and one-colored wallpaper

patterns (chapter 4); but I prefer a purely non-graphical description of one-

colored patterns in order to test/develop the students' reading skills a bit!

Does Isometrica discriminate against glide reflection?

How did you know? You must have read the entire book! Yes, there is some

discrimination ... in the sense that glide reflection is viewed as an isometry

'weaker' than reflection. This view is of course dictated by the fact that glide

reflection, which may certainly be viewed as deferred reflection, is harder to

detect in a wallpaper (or border) pattern. Further, every wallpaper pattern

reflection generates translation(s) parallel to it and, therefore, "hidden glide

reflection(s)": reflection 'contains' glide reflection, but not vice versa (and despite

the fact that every reflection may be viewed as a glide reflection the gliding vector

of which has length zero). But a careful reading of section 8.1 shows that

reflection and glide reflection are simply two equivalent 'possibilities'; and the

'shifting' processes introduced in sections 4.2 - 4.4 clearly indicate that reflection

is the exception that verifies the rule (glide reflection).

One way or another, the teacher must stress the curious interplay between

reflection and glide reflection outlined above, and also insist that the students use

dotted (read dashed) lines for glide reflection axes and vectors and solid lines for

reflection axes and translation vectors, as in the symmetry plans. (There are

places in Isometrica where some readers may disagree with my choice of solid

or dotted lines; when a pattern reflection is combined with a parallel translation in

order to create a 'hidden' glide reflection, for example, I use solid rather than

dotted lines.)

xiv

What is the role of inconsistency with color?

Between the 'perfectly symmetrical' two-colored patterns of Washburn & Crowe

and the randomly colored designs of the 'real world' lies a third, somewhat

esoteric, class of two-colored patterns where, informally speaking, there is some

order within their coloring disorders; more formally, some of their isometries

happen to be inconsistent with color -- reversing colors in some instances and

preserving colors in other instances -- but, otherwise, the coloring appears to be

perfectly symmetrical, and with the two colors in perfect balance with each other

in particular. Such inconsistently yet symmetrically colored patterns are largely

absent from Washburn & Crowe, and for a good reason: it seems that native

artists, driven perhaps by instinct or intuition, largely shunned them, producing

either 'perfect' or 'random' colorings!

A natural question arises: should such inconsistent colorings be avoided in

teaching? Although I do cover this topic extensively in MAT 103 and Isometrica,

my answer is a reluctant "perhaps" -- especially to those teachers who may think

that two-colored patterns would already strain their students considerably. On the

other hand, anyone delving into this seemingly esoteric topic will be rewarded

with many fascinating (both visually and conceptually) creations; the color

inconsistencies involved will often transform a 'symmetrically rich' structure into a

'lower' type, illustrating the fateful principle that "coloring may only reduce

symmetry". Anyway, those wishing to avoid the topic should be able to do so

relatively easily, despite the presence of several color-inconsistent examples;

and those venturing into it may be seduced enough to substantially enlarge

Isometrica's collection of inconsistent colorings!

xv

What is the role of the Conjugacy Principle?

The Conjugacy Principle states that the image of an isometry by any other

isometry is an isometry of the same kind (with rotation angles or glide reflection

vectors preserved modulo orientation); conversely, any two 'identical-looking'

isometries are actually images of each other under a third isometry. In the

context of wallpaper patterns, the Conjugacy Principle becomes an indispensable

tool for their structural understanding and classification. Although formally

introduced in section 6.4 (with the excuse of understanding the color effect of

coexisting reflections and glide reflections) and applied throughout chapter 6, the

Conjugacy Principle is thoroughly discussed and rigorously explained only in

section 8.0 (paving the way for the classification of wallpaper patterns); it also

appears in section 4.0 -- to the extent needed for the establishment of the

Crystallographic Restriction (on rotation angles allowed for wallpaper patterns),

which could admittedly wait until section 8.0.

What do we make of chapter 8?

This final chapter is devoted to my purely geometrical argument that there exist

precisely 17 types of wallpaper patterns. It would clearly be beyond the scope of

most General Education courses, and probably too sophisticated for the great

majority of non-science majors as well. But it is largely self-contained -- totally

self-contained in case section 4.0 and chapter 7 are assumed -- and requires

mathematical maturity rather than knowledge. Interested instructors (or other

readers) should probably teach/read it in parallel with Crystallography Now, a

web page (http://www.oswego.edu/~baloglou/103/seventeen.html) devoted

to a more informal presentation of my classification of wallpaper patterns.

xvi

Comments for experts

Does chapter 8 really offer a classification of wallpaper patterns?

Tough question! The answer depends even on the way one defines a wallpaper

pattern, and whether one believes that Group Theory has to be part of that

definition in particular. Among thousands of visitors of Crystallography Now,

only one was kind enough to tell me that my classification is "more intuitive than

others, but not at all rigorous", his main point being that "two wallpaper patterns

are of the same type if and only if their isometry groups are isomorphic". Fair

enough, but is it reasonable to be able to characterize such simple structures,

known to humankind for thousands of years, only in terms of advanced

mathematical concepts? How would Euclid describe -- and perhaps classify -- the

seventeen types in the Elements, had he included them there? (Just a thought!)

To be honest, a solid structural understanding of the seventeen types of

wallpaper patterns was, and still is, more important to me than a rigorous/quick

proof that there exist indeed precisely seventeen such types. Nonetheless, I

suspect that what Isometrica offers could easily be turned into a formal proof by

replacing isomorphism of isometry groups by a properly defined 'isomorphism' of

symmetry plans. Such an isomorphism would certainly distinguish between solid

lines (reflection) and dotted lines (glide reflection) or between hexagonal dots

(sixfold centers) and triangular dots (threefold centers), etc. Under such an

approach, any two symmetry plans consisting only of round dots (half turn

centers) should represent the same type of wallpaper pattern (p2); even more

frighteningly, any two wallpaper patterns having nothing but translations would be

of the same type (p1) on account of their 'blank' symmetry plans, and so on.

xvii

More interestingly, the reader is invited to compare the way this symmetry plan

approach distinguishes between p4g and p4m (section 8.3) or between p31m

and p3m1 (section 8.4) to the way the traditional group-theoretic approach

reaches the same goals: rather than looking at their generator equations,

Isometrica focuses on the two possible ways in which their (glide) reflections

may 'pass through" their lattices of rotation centers.

[Note: the classification of border patterns in chapter 2 is even more 'informal'

than that of wallpaper patterns, consistently with that chapter's introductory

nature; the interested reader should be able to easily derive a more rigorous

classification of border patterns based on symmetry plans.]

Any new ideas in the proposed classification?

The main new idea is the reduction of complex (rotation + (glide) reflection) types

to the three rotationless types with (glide) reflection (pg, pm, cm) via the

characterization of the latter in terms of their translations. So section 8.1, where

the said characterization is achieved, may seem endless, but the derivation of the

remaining types in the subsequent sections is swift and rather elegant (I hope).

Needless to say, the Conjugacy Principle shines throughout the classification!

Any other surprises prior to chapter 8?

Some readers may find a few interesting ideas lurking in my novel (non-group-

theoretic) classification of two-colored patterns (which assumes the classification

of one-colored patterns), and in the exploitation of symmetry plans in sections 6.9

and 6.11 - 6.12 in particular. Others may be delighted at the various ways of

xviii

passing from one border or wallpaper type to another: although such

'transformations' are included in Isometrica mostly for educational purposes,

they are bold commentaries on the ever-elusive structure of patterns, too!

Can Isometrica's ideas be extended to the three dimensions?

Before trying to explore two-colored 'sparse crystals' (blocks not touching each

other and therefore not obscuring colors) I would rather try to investigate

compositions of three-dimensional isometries in a geometrical context (extending

chapter 7) and classify the 230 crystallographic groups geometrically (extending

chapter 8). I believe that both projects are feasible, and hope to pursue them now

that Isometrica has been completed; anyone interested in competing with me

may like to start with Isometries Come In Circles (my 'mostly two-dimensional'

novel derivation of three-dimensional isometries, currently available at

http://www.oswego.edu/~baloglou/103/circle-isometries.pdf).

What happens when more than two colors are involved?

This question has been answered in Tom Wieting's The Mathematical Theory

of Chromatic Plane Ornaments (Marcel Dekker, 1982). I was ambitious

enough to investigate multicolored types in the context of maplike colorings of

planar tilings, and also without the group-theoretic tools employed by Wieting;

more specifically, I was interested in the interplay between tiling structure and

coloring possibilities. That was not necessarily a hopeless project, and I did/do

have some interesting ideas, but I had to finally admit that my attempts -- during

the summers of 2000 and 2005 -- were not that realistic: several hundred

multicolored tilings later a projected ninth chapter (initially numbered as seventh)

had to be abandoned, and this fascinating, literally colorful, project was

xix

postponed indefinitely... [Section 9.0 (i.e., introduction only) is available at

http://www.oswego.edu/~baloglou/103/isometrica-9.pdf, but has not been

included in Isometrica; it concludes with a 'four color' conjecture on

'symmetrically correct' coloring of tilings.]

Any other future projects related to Isometrica?

It would be nice if someone with more energy and knowledge sits down and

writes a book on wallpaper patterns that could be used for a mathematics

capstone course! Here is how this could be achieved: start with an elementary

geometrical classification of wallpaper patterns like mine and then continue with

the standard group-theoretic classification (available for example in Wieting's

book) and Conway's topological classification, developing/reviewing all needed

mathematical tools along the way. The success of such a project (and course)

would probably depend on the author's ability to delve into the hidden interplay

among the three approaches.

[Conway's orbifold approach may be found, together with broadly related topics,

in Geometry and the Imagination -- informal notes by John Conway, Peter

Doyle, Jane Gilman, and Bill Thurston currently available at

http://www.math.dartmouth.edu/~doyle/docs/gi/gi.pdf; look also for The

Symmetry of Things, by John Conway, Heidi Burgiel, and Chaim Goodman-

Strauss (AK Peters, forthcoming).]

Can we judge this book by its cover?

No way! The figure on the cover is a tribute to the great crystallographer (and not

only) Arthur Loeb and his Color and Symmetry (Wiley, 1971), which offers an

xx

alternative geometrical study of wallpaper patterns. More specifically, it is a

humorous reminder of Loeb's nifty derivation of the composition of two

intersecting glide reflections (and that mysterious parallelogram associated with

them): this important problem forms the pinnacle of my discussion of isometry

composition in chapter 7, and it seems to be absent from all other books that

could have discussed it; my approach is not as direct as Loeb's, but it has its own

methodological advantages (such as requiring a thorough discussion of the

composition of a glide reflection and a rotation, a topic not directly addressed by

Loeb).

[Which Isometrica figure would be on the cover if I didn't choose to attract the

reader's attention to Loeb's work and genius? Tough question, but the winner is

figure 8.19 (on the 'ruling' and unexpected mirroring of half turn centers by glide

reflection): in addition to capturing Isometrica's spirit, it could lead to an

alternative and probably quicker discussion of half turn patterns in section 8.2.

And a close second would no doubt be figure 8.39, which dispenses of the

patterns with threefold/sixfold rotation and reflection by showing that their only

'factor' can be a cm.]

Further comments, acknowledgments, dedications.

Responding to my May 2000 talk at a Madison conference honoring Donald

Crowe, H. S. M. Coxeter -- in his 90's at the time, seated in a wheelchair barely

ten feet from the speaker(s) -- remarked with a wry smile that "all the two-colored

types had been derived in the 1930's by a textile manufacturer from Manchester

[H. J. Woods] without using any Mathematics". The eminent geometer's remark

captures much of the spirit in which Isometrica has been written, as well as the

xxi

subject's precarious position between Art and Mathematics. At another level,

Coxeter's remark serves as a reminder of the interplay and struggle between

rigor and intuition, between structure and freedom, which has certainly left its

mark on Isometrica.

I like to say, in hindsight, that border and wallpaper patterns are "of limited

interest to many people" -- not artistic enough for artists and not mathematical

enough for mathematicians... Further, and contrary to the pleasant illusions

created by Stevens or Washburn & Crowe or Isometrica, symmetry itself is an

exception rather than a rule in the real world: I was rather flattered to hear from

two former students that they think of me when they run across symmetrical

figures during their New York City strolls, but how frequent, and how important

after all, are such symmetrical encounters? How meaningful is abstract beauty in

an increasingly tormented world? I have been caught telling friends that it is not

enough for me to hear my students say that they enjoyed my course (and, by

extension, book), I actually need to hear -- even if occasionally -- that it changed

their life, or, less arrogantly on my part, that “it caused them see the world a little

differently" (this is quoted verbatim from a former student's recent e-mail).

If you read between the lines above you already know that the teaching of MAT

103 and the writing of Isometrica have certainly changed my life: I knew that

since the first week of classes in Fall 1995, when I came up with an assignment

calling for the creation of the seven border pattern types using vertical and

horizontal congruent rectangles -- an assignment that looks trivial now but kept

me up late that night (because the idea of 'multidecked' border patterns is not

'natural' to our minds, perhaps). Moreover, there I was, someone with absolutely

no prior interest in drawing or Design, spending many hours and nights creating

'new' patterns, first by hand, then on a computer ... gradually discovering how

such patterns and concepts could form a gateway to mathematical thought for

students as interested in Mathematics as I once was in Design! [The term

xxii

"design" is used quite narrowly here, and intentionally so: Graphic Design majors

who take MAT 103 tend to find its patterns rather inspiring!]

So a labor of love it was, and this is why I have largely preserved Isometrica in

its original form: perhaps my preferred strategy or tactics for presenting this

incredibly flexible material have changed since 2001, but I chose to preserve my

initial insight and the writing adventure that ensued. For the same reason,

combined with various personal circumstances, Isometrica is going straight to

the internet rather than some constricting publishing house: the software

packages employed (MathWriter and SuperPaint) were already ancient when I

started, the English may seem awkward here and there, the figures are

somewhat primitive and often imperfect, the overall format is kind of kinky, but

you are getting the real thing, and for free at that! [You may in particular get a

good sense of the struggle and discovery process that went on as the exposition

revs up through the chapters: even if there is a "royal road to geometry" … I often

fail to follow it ... keeping in mind that “the shortest approach is not always the

most interesting”!]

My joy at having been able to preserve Isometrica's desired form is offset by the

sadness of having left so much out: my plans of including everything bypassed

by 'first insight' in the form of exercises had to be abandoned, but I am still hoping

of creating additional web pages -- probably linked to the online version of this

Introduction -- in the future, covering extra topics in detail (and color); and if this

hope never materializes, with the future of MAT 103 as inevitably unclear as is, I

trust that enough material has been included here to inspire others toward new

mathematical ideas and/or artistic creations. [Please forgive this desperate

optimism about Isometrica being read and even expanded, but it is my firm

belief that its informal and adventurous style is going to win it some lasting

friends!]

xxiii

My obvious desire to generate disciples for Isometrica has a non-obvious

implication: despite the copyright notices at the beginning and ending of each

chapter, I do allow the reproduction of my book for educational purposes; if for

example a teacher anywhere in the world wishes to have hard copies (of either

Isometrica in its totality or some of its chapters) for his/her students, then it is

fine with me to have that school's printing service produce such copies, even if at

a reasonable cost and marginal profit. So please do not write to me for

permissions (concerning either Isometrica or various web pages related to it): I

would love to have feedback from you, but giving me credit for the materials you

have used is all that I am asking for...

For every book and completed project that sees the light of day there are several

visions buried under perennial darkness: I happen to have the right personality

for incompleteness, therefore I am almost ecstatic as these final lines are being

written; repeatedly seduced as I was by those 'repeating patterns', the discipline

often failed to match the excitement, the time and will appeared not to be there at

times, the questions tended to dwarf the answers... While several friends and

colleagues provided constant support, I believe that the project's completion and,

I hope, success is primarily due to my MAT 103 students and their enthusiasm.

At the risk of being oblivious to the small but precious contributions of many, I

would like to single out and thank five former students for their encouragement

and inspiration: Terry Loretto (Fall 1995), Dreana Stafford (Spring 1999), Michael

Nichols (Fall 1999), who also provided crucial assistance with SuperPaint in

January 2000, Richard Slagle (Fall 2003), and Bonita Bryson (Spring 2004), who

also wrote the aforementioned honors thesis (on the tiling of wallpaper patterns

by border patterns).

As made clear in the beginning of this Introduction, there would simply be no

Isometrica without Margaret Groman's original vision; I am equally grateful to

her for her constant encouragement and suggestions for improvement. Likewise,

xxiv

I am indebted to Mark Elmer, who has also taught MAT 103 several times, for his

careful reading of Isometrica and useful observations. Beyond MAT 103, I am

grateful to my friend and collaborator Phil Tracy, who has also read Isometrica

and discussed it with me in considerable detail; and likewise to my colleagues

Chris Baltus, Fred Barber, Joseph Gaskin, Michel Helfgott, and Kathy Lewis for

their mathematical camaraderie over the years.

Beyond Oswego, I am grateful to a number of mathematicians and others who

provided links to Isometrica's early ambassador, Crystallography Now, or

offered useful feedback: Helmer Aslaksen, Andrew Baker, Dror Bar-Natan, Bryan

Clair, Marshall Cohen, Wis Comfort, David Eppstein (Geometry Junkyard),

Sarah Glaz, Andreas Hatzipolakis, Dean Henderson, William Huff, Loukas

Kanakis, Nikos Kastanis, Barbara Pickett, Doug Ravenel, Jim Reid, Saul Stahl,

Tohsuke Urabe, Marion Walter, Eric Weisstein (Wolfram MathWorld), Mark

Yates, and others – notably family and friends in Thessaloniki, contributors to the

sci.math newsgroup, and participants of my January 2003 Symmetry For All

MAA minicourse -- who should forgive me for having overlooked their input. I am

also grateful to George Anastassiou, Varoujan Bedros, and Fred Linton for their

advice on technical and !legal" matters; along these lines, special thanks are also

due to my friend and non-mathematical collaborator Nick Nicholas.

Back to Oswego, I am grateful to Alok Kumar, Ampalavanar Nanthakumar, and

Bill Noun for their support and good advice; same applies to several other

colleagues from Mathematics, Computer Science, Art and other departments

(and also administration) at SUNY Oswego. Sue Fettes deserves special mention

for her assistance with MathWriter (in its final years). Finally, many thanks are

due to Patrick Murphy, Jean Chambers & David Vampola, and Julia & Matthew

Friday for many a pleasant evening -- followed at times by all-night Isometrica

writing and, inevitably, drawing -- in tranquil Oswego.

xxv

In a somber tone now ... even though Isometrica was dedicated from the

beginning to the memory of our colleague Ron Prisco (Margaret Groman's

Abstract Algebra teacher forty years ago, among other things), who passed away

before even I started writing it but "had a lot of faith in my work", I would like to

honor here the memory of a few local friends whom we lost during the last couple

of years:

-- Bob Deming, whose unpublished but highly effective notes on Linear

Programming provided an early model for me on classroom-generated books

-- Jim Burling, who also taught MAT 103 a couple of times, organized our

seminar, and was a fatherly figure for a number of younger colleagues

-- Gaunce Lewis (of Syracuse University), whose tragically untimely death was a

haunting reminder of the fragility of intellectual pursuits

-- Don Michaels, who in his capacity as tireless news & web administrator

contributed handsomely to the success of MAT 103

Finally, Isometrica owes a lot to my late father, Christos Baloglou (1919 - 2002):

a high school geometer who also taught Descriptive & Projective Geometry to

Aristotle University Engineering students in the 1960's and published Scattered

Drops of Geometry in 2001, he certainly influenced me to study Mathematics.

My whole symmetry project may be seen as a Sisyphean effort to annul his

lovely -- and, less obviously, loving -- verdict on it: "Son, this is not Mathematics"!

George Baloglou

Oswego, April 27, 2007

Table of Contents

CHAPTER 1: ISOMETRIES AS FUNCTIONS

1.0 Functions and isometries on the plane 1 1.1 Translation 7

1.2 Reflection 11

1.3 Rotation 17

1.4 Glide reflection 27

1.5* Why only four planar isometries? 36

CHAPTER 2: BORDER PATTERNS

2.0 Infinity and Repetition 43

2.1 Translation left alone (p111) 44

2.2 Mirrors galore (pm11) 46

2.3 Only one mirror (p1m1) 48

2.4 Footsteps (p1a1) 51

2.5 Flipovers (p112) 54 2.6 Roundtrip footsteps (pma2) 58

2.7 A couple’s roundtrip footsteps (pmm2) 63

2.8 Why only seven border patterns? 66

2.9 Across borders 68

CHAPTER 3: WHICH ISOMETRIES DO IT?

3.0 Congruent sets 77

3.1 Points 82

3.2 Segments 86

3.3 Triangles 90

3.4 Isosceles triangles 96

3.5 Parallelograms, “windmills” and Cn sets 99

3.6 Rhombuses, “daisies”, and Dn sets 105

3.7* Cyclic (Cn) and dihedral (Dn) groups 110

CHAPTER 4: WALLPAPER PATTERNS

4.0 The crystallographic restriction 116

4.1 3600, translations only (p1) 131

4.2 3600 with reflection (pm) 134

4.3 3600 with glide reflection (pg) 137

4.4 3600 with reflection and glide reflection (cm) 141

4.5 1800, translations only (p2) 146

4.6 1800, reflection in two directions (pmm) 149

4.7 1800, reflection in one direction with perpendicular glide reflection (pmg) 151

4.8 1800, glide reflection in two directions (pgg) 154

4.9 1800, reflection in two directions with in-between glide reflections (cmm) 157

4.10 900, four reflections, two glide reflections (p4m) 161

4.11 900, two reflections, four glide reflections (p4g) 165

4.12 900, translations only (p4) 169

4.13 600, six reflections, six glide reflections (p6m) 171

4.14 600, translations only (p6) 174

4.15 1200, translations only (p3) 175

4.16 1200, three reflections, three glide reflections, some rotation centers off reflection axes (p31m) 178

4.17 1200, three reflections, three glide reflections, all rotation centers on reflection axes (p3m1) 180

4.18 The seventeen wallpaper patterns in brief (summary) 183

CHAPTER 5: TWO-COLORED BORDER PATTERNS

5.0 Color and colorings 186

5.1 Colorings of p111 191

5.2 Colorings of pm11 193

5.3 Colorings of p1m1 196

5.4 Colorings of p1a1 200

5.5 Colorings of p112 202

5.6 Colorings of pma2 204

5.7 Colorings of pmm2 206

5.8 Consistency with color 211

5.9 Symmetry plans 215

CHAPTER 6: TWO-COLORED WALLPAPER PATTERNS

6.0 Business as usual? 220

6.1 p1 types (p1 , pb′ 1) 225

6.2 pg types (pg, pb′ 1g, pg′) 227

6.3 pm types (pm , pm ′, pb′ 1m , p ′m , pb′ g, c′m) 233

6.4 cm types (cm, cm ′, pc′ g, pc′ m) 242

6.5 p2 types (p2 , p2 ′, pb′ 2) 252

6.6 pgg types (pgg, pgg′, pg′g′) 257

6.7 pmg types (pmg, pb′ mg, pmg′, pm′g, pb′ gg, pm′g′) 266

6.8 pmm types (pmm , pb′ mm , pmm ′, c′mm , pb′ gm, pm ′m ′) 273

6.9 cmm types (cmm, cmm ′, cm ′m ′, pc′ mm , pc′ mg, pc′ gg) 281

6.10 p4 types (p4 , p4 ′, pc′ 4) 294

6.11 p4g types (p4g, p4′gm ′, p4′g′m , p4g′m ′) 298

6.12 p4m types (p4m , p4 ′mm ′, pc′ 4mm , pc′ 4gm , p4 ′m ′m , p4m ′m ′) 303

6.13 p3 types (p3) 312

6.14 p31m types (p31m, p31m′) 313

6.15 p3m1 types (p3m1, p3m ′) 316

6.16 p6 types (p6, p6′) 320

6.17 p6m types (p6m , p6mm ′, p6 ′m ′m , p6 ′m ′m ′) 324

6.18 All sixty three types together (symmetry plans) 332

CHAPTER 7: COMPOSITIONS OF ISOMETRIES

7.0 Isometry ‘hunting’ 341

7.1 Translation ∗ Translation 350

7.2 Reflection ∗ Reflection 351

7.3 Translation ∗ Reflection 354

7.4 Translation ∗ Glide Reflection 356

7.5 Rotation ∗ Rotation 359

7.6 Translation ∗ Rotation 366

7.7 Rotation ∗ Reflection 372

7.8 Rotation ∗ Glide Reflection 376

7.9 Reflection ∗ Glide Reflection 380

7.10 Glide reflection ∗ Glide Reflection 385

CHAPTER 8: WHY PRECISELY SEVENTEEN TYPES?

8.0 Classification of wallpaper patterns 393

8.1 3600 patterns 398

8.2 1800 patterns 413

8.3 900 patterns 425

8.4 1200 and 600 patterns 429

Basic Index

border pattern 2.0.1 [43]

border pattern ‘backbone’ 2.3.2 [49]

circular orientation 1.5.4 [42]

clockwise 1.3.1 [18]

Cn sets 3.5.4 [104]

color preservation 5.0.3 [188]

color reversal 5.0.3 [187]

commuting (isometries) 1.4.2 [28]

congruent sets 3.0.1 [77]

Conjugacy Principle 4.0.5 [124]

conjugate 4.0.4 [121]

consistency with color 5.8.3 [213]

consistent coloring 7.0.3 [342]

coordinates 1.0.2 [1]

counterclockwise 1.3.3 [19]

Crystallographic Restriction 4.0.3 [120]

crystallographic notation 5.3.1 [198]

cyclic group 3.7.1 [112]

dihedral group 3.7.2 [115]

distance formula 1.5.1 [36]

Dn sets 3.6.3 [109]

fundamental region 2.1.2 [45]

glide reflection 1.4.2 [28]

half turn 1.3.10 [27]

heterostrophic sets 3.0.2 [79]

hidden glide reflection 2.7.1 [64]

homostrophic sets 3.0.2 [79]

image 1.0.1 [1]

in-between glide reflection 4.4.3 [144]

inconsistent with color 5.8.1 [211]

inverse 1.4.4 [30]

isometry 1.0.6 [4]

isometry ‘product’ (composition) 4.04 [121]

isometry ‘weaving’ 6.9.3 [287]

labeling 3.0.3 [79]

lattice (of rotation centers) 4.0.4 [122]

linear function 1.5.1 [36]

map(ping) 1.0.1 [1]

maplike coloring 7.0.3 [342]

midpoint 1.2.6 [14]

minimal translation 2.1.1 [45]

mirror symmetry 1.2.8 [17]

n-fold rotation 3.5.4 [104]

one-colored 5.0.1 [186]

parallelogram rule 4.1.1 [132]

parent type (of a two-colored pattern) 5.9.1 [215]

perfect shifting 4.4.2 [142]

perfectly shifted stacking 4.4.1 [141]

perpendicular bisector 3.2.2 [87]

point reflection 1.3.10 [27]

Postulate of Closest Approach 4.0.4 [123]

random shifting 4.4.2 [142]

rectangular ruling (of half turn centers) 4.8.2 [155]

reflection 1.2.2 [11]

rotation 1.3.2 [19]

rotational symmetry 1.3.9 [26]

sense (of a vector) 1.1.5 [11]

shifted stacking 4.3.1 [137]

shifting 2.7.3 [65]

smallest rotation 4.0.3 [119]

smallest rotation consistent with color 6.0.2 [221]

stacking (of border patterns) 4.1.1 [131]

straight stacking 4.2.1 [135]

symmetry plan 5.2.3 [195]

symmetry decrease 5.8.2 [212]

T0 (vector) 8.1.4 [401]

T1 (vector) 8.1.3 [400]

T2 (vector) 8.1.4 [401]

translation 1.1.2 [8]

triangle inequality 1.0.7 [5]

two-colored 5.0.2 [187]

vector-angle 1.1.5 [10]

wallpaper pattern 4.0.1 [116]

2006 George Baloglou first draft: summer 1998

CHAPTER 1

ISOMETRIES AS FUNCTIONS

1.0 Functions and isometries on the plane

1.0.1 Example. Consider a fixed point O and the following ‘operation’: given any other point P on the plane, we send (map) it to a point P′ that lies on the ray going from O to P and also satisfies the equation |OP ′′′′| = 3 ×××× |OP| (figure 1.1). It is clear that for each point P there is precisely one point P′, the image of P, that satisfies the two conditions stated above. Any such process that associates precisely one image point to every point on the plane is called a function (or mapping) .

Fig. 1.1

1.0.2 Coordinates. Let us now describe the ‘blowing out’ function discussed in 1.0.1 in a different way, using the cartesian coordinate system and positioning O at the origin, (0, 0). Consider a specific point P with coordinates (2.5, 1.8). Looking at

the similar triangles OPA and OP′B in figure 1.2, we see that |P′B||PA|

= |OB||OA|

= |OP′||OP|

= 3, hence |P′B| = 3 × |PA| = 3 × 1.8 = 5.4 and |OB| = 3 × |OA|

1

= 3 × 2.5 = 7.5. That is, the coordinates of P′ are (7.5, 5.4). In exactly the same way we can show that an arbitrary point with coordinates (x, y) is mapped to a point with coordinates (3x, 3y). We may therefore represent our function by a formula: f(x, y) = (3x, 3y).

Fig. 1.2

1.0.3 Images. Let us look at the rectangle ABCD, defined by the four points A = (2, 1), B = (2, 2), C = (5, 2), D = (5, 1). What happens to it under our function? Well, it is simply mapped to a ‘blown out’ rectangle A′B ′C ′D ′ -- image of ABCD under the ‘blow out’ function --with vertices (6, 3), (6, 6), (15, 6), and (15, 3), respectively:

Fig. 1.3

2

1.0.4 More functions. One can have many more functions, formulas, and images. For example, g(x,y) = (2x−−−−y, x+3y) maps ABCD to a parallelogram, while h(x, y) = (3x+y, x−−−−y2+4) maps ABCD to a semi-curvilinear quadrilateral (figure 1.4). We compute the images of A under g and h, leaving the other three vertices to you: g(A) = g(2, 1) = (2×2−1, 2+3×1) = (3, 5); h(A) = h(2, 1) = (3×2+1, 2−12+4) = (7, 5). (You may find more details in 1.0.8.)

Fig. 1.4

1.0.5 Distortion and preservation. Looking at the three functions f, g, and h we have considered so far, we notice a progressive ‘deterioration’: f simply failed to preserve distances (mapping ABCD to a bigger rectangle), g failed to preserve right angles (but at least sent parallel lines to parallel lines), while h did not even preserve straight lines (it mapped AB and CD to curvy lines). Now that we have seen how ‘bad’ some (in fact most) functions can be, we may as well ask how ‘good’ they can get: are there any functions that preserve distances (therefore angles and shapes as well), satisfying |AB| = |A′′′′B ′′′′| for every two points A, B on the plane?

3

The answer is “yes”. Distance-preserving functions on the plane do exist, and we can even tell exactly what they look like: they are defined by formulas like F(x, y) = (a′′′′+b ′′′′x+c ′′′′y, d′′′′+e ′′′′x+f ′′′′y), where a ′, d′ are arbitrary, b′2+e ′2 = c′2+f′2 = 1, and either f′ = b′, e′ = −c′ or f′ = −b ′, e′ = c′! This is quite a strong claim, isn’t it? Well, we will spend the rest of the chapter proving it, placing at the same time considerable emphasis on a geometric description of the involved functions. For the time being you may like to check what happens when b′ or c′ are equal to 0: what is the image of ABCD in these cases? (Look at specific examples involving situations like a′ = 3, b′ = 0, c′ = 1, d′ = 2, e′ = -1, f′ = 0 or a′ = -2, b′ = -1, c′ = 0, d′ = 4, e′ = 0, f′ = 1, and determine the images of A, B, C, D.)

1.0.6 What’s in a name? You probably feel by now that such nice, distance and shape preserving functions like the ones mentioned above deserve to have a name of their own, don’t you? Well, that name does exist and is probably Greek to you: isometry, from ison = “equal” and metron = “measure”. The second term also lies at the root of “symmetry” = “syn” + “metron” = “plus” + “measure” (perfect measure, total harmony). In fact ancient Greek isometria simply meant “symmetry” or “equality”, just like the older and more prevalent symmetria . The term “isometry” with the meaning “distance-preserving function” entered English -- emulating somewhat earlier usage in French and German -- in 1941, with the publication of Birkhoff & Maclane’s Survey of Modern Algebra.

1.0.7 Isometries preserve straight lines! We claim that every isometry maps a straight line to a straight line. And, yes, one could prove this claim without even knowing (yet) what isometries look like, without having seen a single example of an isometry! In fact, one could prove that isometries preserve straight lines without even knowing for sure that isometries do exist!! This mathematical world can at times be a strange one, can’t it? But how do we prove such an ‘abstract’ claim?

Well, a clever observation is crucial here: it suffices to show that every isometry maps three distinct collinear points to three

4

distinct collinear points! Indeed, let’s assume this ‘subclaim’ for now, and let’s prove right below the following: every function (not necessarily an isometry!) that maps every three collinear points to three collinear points must also map every straight line L to (a subset of) a straight line L′. Once this is done, preservation of distances shows easily that the image of L actually ‘fills’ L′.

Start with a straight line L and pick any two distinct points P1, P2 on it. These two points are mapped by our function to distinct points P1′ , P2′ that certainly define a new line, call it L′. Now every other point P on L is collinear with P1, P2, therefore, by our subclaim above (still to be proven!), its image P′ is collinear with P1′ , P2′ , hence it lies on L′ (figure 1.5). That is, every point P on L is mapped to a point P′ on L′, hence L itself is mapped ‘inside’ L′.

Fig. 1.5

So, how do we prove our subclaim that an isometry must always map collinear points to collinear points? Well, let A, B, C be three collinear points that are mapped to not necessarily collinear points A ′, B′, C′, respectively (figure 1.6). We are dealing with an isometry, therefore |A′C′| = |AC|, |A′B′| = |AB|, and |B′C′| = |BC|. Since A, B, C are collinear, |AC| = |AB| + |BC|. But then |A′C′| = |AC| = |AB| + |BC| = |A′B′| + |B′C′|. We are forced to conclude that A′, B′, C′ must indeed be collinear: otherwise one side of the triangle A′B ′C ′ would be equal to the sum of the other two sides, violating the familiar triangle inequal i ty .

5

Fig. 1.6

1.0.8 Practical (drawing) issues. As you will experience in the coming sections, the fact that isometries map straight lines to straight lines makes life a whole lot easier: to draw the image of a straight line segment, for example, all you have to do is determine the images of the two endpoints and then connect them with a straight line segment. On our part, we will be repeatedly applying this principle throughout this chapter without specifically mentioning it.

On the other hand, determining the image of either a straight segment under a function that is not an isometry or a curvy segment under any function requires more work: one needs to determine the images of several points between the two endpoints and then connect them with a rough sketch. This is, for example, how h(AB) has been determined in 1.0.4: h(2, 1) = (7, 5), h(2, 1.2) = (7.2, 4.56), h(2, 1.4) = (7.4, 4.04), h(2, 1.6) = (7.6, 3.44), h(2, 1.8) = (7.8, 2.76), h(2, 2) = (8, 2). This is indeed a lot of work, especially when not done on a computer! Luckily, most images in this book are determined geometrically rather than algebraically; more to the point, most shapes under consideration will be quite simple geometrically, defined by straight lines.

1.0.9* How about parallel lines? Now that you have seen why isometries must map straight lines to straight lines, could you go one step further and prove that isometries must also map parallel lines to parallel lines? You can do this arguing by contradiction: suppose that parallel lines L1, L2 are mapped by an isometry to non-parallel lines L1′ , L2′ intersecting each other at point K; can you then

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notice something impossible that happened to those distinct points K1, K2 (on L1, L2, respectively) that got mapped to K?

And if you are truly adventurous, can you prove, perhaps by contradiction, that whenever a function (not necessarily an isometry!) maps straight lines to straight lines it must also map parallel lines to parallel lines?

1.1 Translation

1.1.1 Example. Consider the triangles ABC, A′B′C ′ below:

Fig. 1.7

Not only they are congruent to each other, but they also happen to be ‘parallel’ to each other: AB, BC, and CA are parallel to A′B ′, B ′C ′, and C′A ′, respectively. This is a rather special situation, and what lies behind it is a vector .

1.1.2 Vectors. Familiar as it might be from Physics, a vector is a hard-to-define entity. It basically stands for a uniform motion that takes place all over the plane: every single point moves in the same direction (the vector’s direction -- but have a look at 1.1.5, too) and by the same distance (the vector’s length). In figure 1.7, for example, it is easy to see that every point of the triangle ABC has moved in the same southwest to northeast (SW-NE) direction by

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the same distance. We represent this motion by the ‘arrow’ below and call it a translation -- “transferring (ABC) to (A′B ′C ′)”, in the same way a text is transferred from one language to another --

defined by the vector v→→→→

:

Fig. 1.8

Comparing figures 1.7 and 1.8, we easily conclude that every vector uniquely defines a translation and vice-versa. Notice also that the triangle A′B ′C ′ moves back to the triangle ABC by a translation opposite of the SW-NE one we already discussed, a translation defined by a NE-SW vector of equal length:

Fig. 1.9

1.1.3 It’s an isometry! While figure 1.7 makes it ‘obvious’ that every translation does preserve distances, it would be nice to actually have a proof of this claim. All we need to do is to show that

if points A and B move by the same vector v→

to image points A′, B′ then |A′′′′B ′′′′| = |AB|. But this is easy: as AA′ and BB′ are ‘by definition’ parallel and equal to each other, AB and A′B ′ are by necessity the opposite, therefore equal (and parallel), sides of a parallelogram:

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Fig. 1.10

1.1.4 Coordinates. Let us revisit 1.1.1, placing now figure 1.7 in a cartesian coordinate system, so that the coordinates of A, B, C and the approximate coordinates of A′, B′, C′ are as shown below:

Fig. 1.11

It doesn’t take long to realize that our translation simply adds approximately 3.6 units to the x-coordinate of every point and approximately 2.5 units to the y-coordinate of every point; this is explicitly shown in figure 1.11 for C and C′. We call these two numbers coordinates of the translation vector, which we may now

write as v→→→→

≈≈≈≈ . By the Pythagorean Theorem , the

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vector’s length is approximately 3.62 + 2.52 ≈ 4.3. All this is further clarified in figure 1.12, which in particular shows how the translation vector’s coordinates are determined by the image O′ of (0, 0):

Fig. 1.12

That is, we may represent this translation employing the formula T(x, y) = (3.6+x, 2.5+y). More generally, every translation on the plane may be represented by a formula of the form T(x, y) = (a+x, b+y). Conversely, each formula of the form T(x, y) = (a+x, b+y) represents a translation defined by the vector ; sometimes we may even denote the translation itself by . Observe that the opposite of the translation defined by the vector is simply defined by the vector . For example, the opposite translation of that we discussed in 1.1.2 is .

1.1.5 ‘Determining’ a vector. While figure 1.12 provides sufficient illustration on the relation between a vector’s length, direction, and coordinates, just a bit of Trigonometry makes

everything so much clearer! Indeed, since the vector v→

of length 4.3 makes a vector-angle of about 360 with the positive x-axis, its x-coordinate and y-coordinate are given by 4.3 × cos360 ≈ 4.3 × .81 ≈ 3.48 and 4.3 × sin360 ≈ 4.3 × .59 ≈ 2.53, respectively. While absolute precision has not been achieved, is indeed very close to . The quotient 2.53/3.48 ≈ .73 is the vector’s slope , which is another quantitative way of describing the vector’s direction. (Those who know a bit more know of course that this slope is equal to approximately tan360.)

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Beware at this point of a simple, yet important, fact: while the two distinct vectors and share the same direction, their slopes being equal (2.5/3.6 = (−2.5)/(−3.6)), they are of opposite sense, going opposite ways (as figures 1.8 & 1.9 demonstrate). Moreover, as we shall see in 1.4.7, opposite vectors have distinct vector-angles, in this case 360 and 2160, respectively. So, it is important to remember that for every slope/direction there exist two distinct, and opposite of each other, senses. Notice at this point that any two vectors of equal length and same direction and sense are one and the same, while any two vectors of equal length and same direction might be either one and the same or opposite of each other.

1.2 Reflection

1.2.1 Mirrors create equals. Anyone who has ever successfully looked into a mirror is aware of this simple, as well as deep, natural phenomenon. Moreover, the closer you stand to a mirror, the closer you see your image in it -- another simple truth that even your cat is likely to be painfully aware of! In fact your mirror image lies precisely as far ‘inside’ the mirror as far away from it you stand: a fact used by many restaurants, bars, etc, to ‘double’ their perceived space. As mirrors or calm ponds cannot be included in books, we need a more abstract way of illustrating such natural observations, and we must indeed invent a ‘paper equivalent’ of a mirror!

1.2.2 Reflection axes. In order to ‘touch’ your mirror image inside a mirror you need to extend your hand toward the mirror straight ahead , so that it makes a right angle with the mirror, right? Well, this simple observation, together with the ones made in 1.2.1, helps us come up with the needed representation of a mirror on paper. The image P′ of a point P under reflection about the axis (mirror) L is found as figure 1.13 indicates:

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Fig. 1.13

That is, we get the same effect on P as if an actual mirror had somehow been placed perpendicularly to this page along L: you may of course go through such an experiment and see what happens!

1.2.3 Images. Let us return to triangle ABC of figure 1.7 and try to find its image under reflection by the straight line L in figure 1.14. We do that simply by determining the images A′, B′, C′ of vertices A, B, C and then connecting them to obtain the image triangle A′B′C ′:

Fig. 1.14

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1.2.4 It’s an isometry! The two triangles in figure 1.14 certainly look congruent. This might not be as obvious as it was in the case of the two triangles of figure 1.7 -- we will elaborate on this in section 3.0 -- but, having three pairs of seemingly equal sides, ABC and A′B′C′ have to be congruent. How do we show that |BC| = |B′′′′C′′′′|, for example?

Fig. 1.15

Well, all we need to do is draw segments CD, C′D ′ (both parallel to L and perpendicular to BB′, CC′) and notice, with the help of the rectangles FECD and FEC′D′ (figure 1.15), that |DF| = |CE| = |C′E| = |D′F|, therefore |DB| = |BF| − |DF| = |B′F| − |D′F| = |D′B′|, while |DC| = |FE| = |D′C ′|: it follows that the two right triangles DBC and D′B ′C ′ are congruent (because |DB| = |D′B′| and |DC| = |D′C′|), hence |BC| = |B′C′|.

1.2.5 Coordinates. Let us now place triangles ABC, A′B ′C ′ and the axis L in a cartesian coordinate system (figure 1.16) and see what happens! You may use your straightedge to estimate the coordinates of A′, B′, and C′ and verify that (2, 1), (2, 3), and (6, 2) got mapped to approximately (5.1, 7.7), (3.6, 6.4), and (6.9, 4), respectively.

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Fig. 1.16

Unlike in the case of translation, there is no obvious algebraic way of describing the transformation of coordinates observed above. This is of course no reason for giving up on determining the magic formula, if any, that lies behind this transformation of coordinates.

1.2.6 The reflection formula. Let L be a straight line with equation ax + by = c and M(x, y) be the mirror image of an arbitrary point (x, y) under reflection about L. Then (‘magic formula’)

M(x, y) =

= ( 2 a ca2+b2

+ b2−−−−a2

a2+b2x −−−− 2 a b

a2+b2y, 2 b c

a2+b2 −−−− 2 a b

a2+b2x −−−− b

2−−−−a2

a2+b2y )

Proof*: Let (x′′′′, y′′′′) be the coordinates of M(x, y) and (x1, y1) be

the coordinates of the midpoint Q of the segment connecting (x, y) and (x′, y′); Q lies, of course, on the mirror L (figure 1.17), while

x1 = x+x′

2 and y1 =

y+y′2

.

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Fig. 1.17

Since (x1, y1) = (x+x′

2, y+y′

2) lies on the line ax + by = c, we

obtain a( x+x′2

)+b( y+y′2

) = c, therefore a(x+x′)+b(y+y′) = 2c,

ax+ax′+by+by′ = 2c, and, finally, ax′′′′+by′′′′ = 2c−−−−ax−−−−by (I).

Next, observe that the line a